A Note on Letters of Yangian Invariants · 2020-07-06 · Prepared for submission to JHEP A Note on Letters of Yangian Invariants Song He a;bZhenjie Li aCAS Key Laboratory of Theoretical

Prepared for submission to JHEP

A Note on Letters of Yangian Invariants

Song Hea,b Zhenjie Lia,b

aCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese

Academy of Sciences, Beijing 100190, ChinabSchool of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan

Road, Beijing 100049, China

E-mail: [email protected], [email protected]

Abstract: Motivated by reformulating Yangian invariants in planar N = 4 SYM

directly as d log forms on momentum-twistor space, we propose a purely algebraic

problem of determining the arguments of the d log’s, which we call “letters”, for any

Yangian invariant. These are functions of momentum twistors Z’s, given by the posi-

tive coordinates α’s of parametrizations of the matrix C(α), evaluated on the support

of polynomial equations C(α) · Z = 0. We provide evidence that the letters of Yan-

gian invariants are related to the cluster algebra of Grassmannian G(4, n), which is

relevant for the symbol alphabet of n-point scattering amplitudes. For n = 6, 7, the

collection of letters for all Yangian invariants contains the cluster A coordinates of

G(4, n). We determine algebraic letters of Yangian invariant associated with any

“four-mass” box, which for n = 8 reproduce the 18 multiplicative-independent, alge-

braic symbol letters discovered recently for two-loop amplitudes.

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Contents

1 Introduction 1

2 Letters of Yangian invariants 4

2.1 Letters of NMHV and MHV invariants 4

2.2 Letters of n = 6 and n = 7 invariants 6

2.3 Algebraic letters of N2MHV invariants 9

3 Discussions 11

1 Introduction

Recent years have witnessed tremendous progress in unravelling hidden mathemati-

cal structures of scattering amplitudes, especially in planar N = 4 supersymmetric

Yang-Mills theory (SYM) (c.f. [1, 2]). Moreover, formidable progress has been made

in computing multi-loop scattering amplitudes of the theory, most notably by the

hexagon and heptagon bootstrap program [3–8]. The first non-trivial amplitude

in planar N = 4 SYM, the six-point amplitude (or hexagon) has been determined

through seven and six loops for MHV and NMHV cases respectively [9], and similarly

the seven-point amplitude (or heptagon) has been determined through four loops for

these cases respectively [10, 11]. A crucial assumption for the hexagon and heptagon

bootstrap is that the collection of the letters entering the symbol (which is the max-

imally iterated coproduct of generalized polylogarithms), or the alphabet, consists of

only 9 and 42 variables known as cluster X coordinates of G(4, n) [12]. These are dual

conformally invariant (DCI), rational functions of momentum twistors describing the

kinematics of n-point amplitudes [13]. The space of generalized polylogarithms with

correct alphabet is remarkably small, and certain information from physical limits

and symmetries suffices to fix the result (see [14] for a recent review).

Starting n = 8, the cluster bootstrap becomes intractable since the Grassman-

nian cluster algebra [15] becomes infinite type. It is an important open question how

to find a finite subset of cluster variables which appear in the symbol alphabet of

multi-loop amplitudes, and there has been significant progress e.g. by studying their

Landau singularities [16–19]. Moreover, it is known that starting n = 8, algebraic

(irrational) functions of momentum twistors appear in the alphabet, as one can al-

ready see in four-mass box integrals needed for one-loop amplitudes with k ≥ 2 (such

algebraic letters are also present for higher-loop MHV amplitudes with n ≥ 8).

– 1 –

Recently, a new computation for two-loop n = 8 NMHV amplitude is performed

using the so-called Q equations [20], which provides a candidate of the symbol alpha-

bet for n = 8; in particular, it provides the space of 18 algebraic letters which involve

square roots of the kinematics. After that new proposals concerning the symbol al-

phabet for n ≥ 8 have appeared [21–23], which exploit mathematical structures such

as tropical Grassmannian [24] and positive configuration space [25].

In this short note, we make a simple observation that an alternative construction,

based on solving polynomial equations associated with plabic graphs for Yangian

invariants, may provide another route to symbol alphabet including algebraic letters.

We will argue that the alphabet of Yangian invariants, which will be defined shortly,

contains not only the (rational) symbol letters of n = 6 and n = 7 amplitudes but

also the algebraic letters for two-loop n = 8 NMHV amplitude mentioned above.

We emphasize that generally for any Yangian invariant, the alphabet includes

but is not restricted to the collection of poles, which have been explored in the

literature so far, e.g. in the context of the so-called cluster adjacency [11, 26–30].

As we will see already for n = 7, by going through all Yangian invariants we find

42 poles in total, which are 7 frozen variables and 35 unfrozen ones, i.e. 7 of the

42 cluster A variables of G(4, 7) are still missing! On the other hand, the alphabet

of all possible Yangian invariants consists of 63 letters, which include the 42 cluster

variables. The search for alphabets of Yangian invariants is by itself a beautiful open

problem related to positive Grassmannian, plabic graphs and cluster algebra [1], and

let’s first formulate the general problem as suggested by [31].

An NkMHV Yangian invariant is an on-shell function of (super-)momentum

twistors {Z|η}1≤a≤n, which is associated with a certain 4k-dimensional positroid

cell of the positive Grassmannian G+(k, n) 1; the cell can be parametrized by a k×nmatrix C({α}1≤i≤4k) where αi’s are positive coordinates, and the on-shell function

(which has Grassmann-degree 4k) is given by

Y ({Z|η}) =

∫ 4k∏i=1

d logαi δ4k|4k(C({α}) · (Z|η)) , (1.1)

where the contour of the integral encloses solutions of equations∑n

a=1CI,a({α})Za =

0 (for I = 1, 2, · · · , k). As first proposed in [32] and studied further in [33], it is in-

structive to write Yangian invariants, or more general on-shell (super-)functions, as

differential forms on momentum-twistor space, where we replace the fermionic vari-

ables ηa by the differential dZa for a = 1, · · · , n. Remarkably, after the replacement

the Yangian invariant defined as a 4k form, Y({Z|dZ}) := Y |ηa→dZa , becomes the

pushforward of the canonical form of the cell,∏

i d logαi, to Z space:

Y∗(Z|dZ) = {4k∧i=1

d logα∗i (Z)}, with {α∗(Z)} solutions of C(α) · Z = 0 . (1.2)

1One can easily extend our construction to general m but in this note we only consider m = 4.

– 2 –

In other words, we evaluate the form on solutions, α∗(Z), of the polynomial equa-

tions C(α) · Z = 0, and we have used subscript ∗ as a reminder that there can be

multiple solutions in which case Y∗ becomes a collection of 4k forms, one for each

solution (the number of solutions, Γ(C) is given by the number of isolated points

in the intersection C⊥ ∩ Z [1]). The solutions, α∗(Z), are GL(1)-invariant functions

of Plucker coordinates on G(4, n), 〈a b c d〉 := det(ZaZbZcZd), and when there are

multiple solutions, Γ(C) > 1, they are generally algebraic functions.

Since symbol letters of loop amplitudes are arguments of generalized polyloga-

rithms, it is only natural to refer to the arguments of the d log’s on the support of

C · Z = 0, as the letters of a given Yangian invariant. One can define the alpha-

bet of a Yangian invariant to be the collection of possible letters, i.e. the solutions

{α∗i (Z)}, which form a 4k × Γ(C) matrix. However, there are ambiguities since

under a reparametrization of the cell, C({α}) → C({α′}), which leaves the form

invariant, the alphabet generally changes. Such reparametrizations are related to

cluster transformations acting on variables of (positroid cells of) G+(k, n) [1]. In

principle one could attempt to find all possible letters for a given positroid cell, by

scanning through all transformations that leave the canonical form invariant, but the

resulting collection of letters can easily be infinite for n ≥ 8 !

Therefore, we restrict ourselves to a very small subset of such transformations,

namely those generated by equivalence moves acting on plabic graphs associated with

a positroid cell. Therefore, we define the alphabet of any given Yangian invariant to

be the collection of face or edge variables of all possible plabic graphs. It is obvious

that this collection of letters is finite for any Yangian invariant, and we can further

take the union of the alphabets for all Yangian invariants of the same n and k, which

will be referred to as the alphabet of a given n and k.

There is still a residual redundancy in this definition: any monomial reparametriza-

tion of the form α′j =∏

i αni,j

i for 1 ≤ j ≤ d with detni,j = 1 leaves the form invariant.

Even for a given plabic graph, we have such redundancy in the definition of face or

edge variables, and the invariant content of the alphabet is the space generated by

monomials of {α∗(Z)i} (or the linear space spanned by {logα∗i }), and one can choose

some representative letters which give the same space. For example, when α′(Z) are

rational with Γ(C) = 1, one can always choose a set of irreducible polynomials

of Plucker coordinates, which are analogous to A coordinates of cluster algebra of

G(4, n). Similar choices can be made for algebraic solutions with Γ(C) > 1, modulo

ambiguities from the fact that monomials of these algebraic letters can form rational

letters which are irreducible polynomials.

Since the letters of Yangian invariants are generally not DCI, there is no unam-

biguous way to extract the analog of cluster X coordinates. However, we will see

that at least for our main example of algebraic letters, where the Yangian invari-

ants correspond to leading singularities of four-mass boxes, there is a natural way

to define such DCI X -like variables, which turns out to give precisely the same 18

– 3 –

algebraic alphabet for two-loop NMHV amplitudes in [34].

2 Letters of Yangian invariants

In the following, we will illustrate our algorithm by finding (subsets of) alphabets of

Yangian invariants for certain n and k. The procedure goes as follows: for any given

n and k, we first scan through all possible 4k-dimensional cell of G+(k, n) and list

the resulting Yangian invariants. The representation of any Yangian invariant can

be obtained using the procedure given in [1], in terms of the matrix C({α}) with

canonical coordinates {α}1≤i≤4k for the cell; equivalently these coordinates can be

identified with non-trivial edge variables of a representation plabic graph. Then one

can start to apply square moves to the graph, which generate reparametrization of

the cell and possibly find more letters. In simple cases, it is straightforward to find

all equivalence moves and obtain the complete alphabet of the Yangian invariant; for

more involved cases, we will content with ourselves in finding a subset of the alphabet

by applying such move once, and the results turn out to be already illuminating.

2.1 Letters of NMHV and MHV invariants

First, we present the simplest non-trivial Yangian invariant, which is the only type

for NMHV (k = 1). A 4-dimensional cell C ∈ G+(1, n) can be parametrized as

(. . . , 0, 1, 0 . . . , 0, α1, 0, . . . , 0, α2, 0, . . . , 0, α3, 0, . . . , 0, . . . , 0, α4, 0, . . . ),

where only a, b, c, d and e-th entry are non-zero, and the on-shell function is given

by the R invariant [a, b, c, d, e]. The solution of the linear equation C ·Z = 0 is simply

α1 = −〈acde〉〈bcde〉

, α2 = −〈abde〉〈cbde〉

, α3 = −〈abce〉〈dbce〉

, α4 = −〈abcd〉〈ebcd〉

, (2.1)

and one can easily check that Yn,1(a, b, c, d, e) =∏4

i=1 d logα∗i (Z) is indeed the form

obtained by replacing η by dZ in [a, b, c, d, e]. Obviously any reparametrization is

trivial in this case, and we can choose representative letters to be the five Plucker

coordinates appeared above.

By taking the union of alphabets of all NMHV invariants for n points, we trivially

get the collection of all Plucker coordinates. Also note that in this special case the

representative letters are exactly the poles of the Yangian invariant, but we will see

immediately that this is no longer true beyond NMHV.

Our next example is for MHV (k = n−4), where the unique Yangian invariant

for n points is given by the top cell of G+(n−4, n) (or equivalently that of G+(4, n),

and the form is the familiar cyclic measure:

Yn,n−4 =d4nZ/(vol GL(4))

(1234)(2345) · · · (n123),

– 4 –

The poles of the invariant (MHV amplitude) are the frozen Plucker coordinates, as

they should be. However, the letters for any plabic graph involve variables other

than these frozen ones. Let’s start with n = 6 (for n = 5 it is just NMHV invariant

[1, 2, 3, 4, 5]), and a representative plabic graph is

1

2

3

4

5

6α3

α2 α4α5

α6 α7α8

α1

The corresponding C matrix reads(1 α2 + α4 + α6 + α8 (α2 + α4 + α6)α7 (α2 + α4)α5 α2α3 0

0 1 α7 α5 α3 α1

)and the solution of C(α) · Z = 0 is

α1 =〈2345〉〈3456〉

, α2 = −〈1234〉〈3456〉〈2345〉〈2346〉

, α3 = −〈2346〉〈3456〉

, α4 = −〈1236〉〈3456〉〈2346〉〈2356〉

,

α5 =〈2356〉〈3456〉

, α6 = −〈1256〉〈3456〉〈2356〉〈2456〉

, α7 = −〈2456〉〈3456〉

, α8 = −〈1456〉〈2456〉

,

where we find the letters given by the following 9 Plucker coordinates:

〈1234〉, 〈2346〉, 〈2345〉, 〈1236〉, 〈2356〉, 〈1256〉, 〈2456〉, 〈1456〉, 〈3456〉. (2.2)

Note that these are the A coordinates of a cluster in G(4, 6): 6 frozen variables plus

3 unfrozen ones, and they can be arranged as the quiver below. This quiver can be

obtained as the dual of the plabic graph [1], and in this case, any square moves on

plabic graphs are in 1:1 correspondence with mutations on the cluster. Thus, without

explicitly performing square moves, we see the complete alphabet of the Yangian is

– 5 –

given by the collection of all A coordinates, which are the 15 Plucker coordinates.

〈2345〉

$$〈2346〉

��

// 〈2356〉

��

// 〈2456〉

��

// 〈3456〉

��

〈1234〉 〈1236〉

dd

〈1256〉

dd

〈1456〉

dd

For general n, one can always choose a representative plabic graph, which cor-

responds to an initial cluster of G(4, n) (as a generalization of the quiver above). It

is straightforward to see that for such a graph, the letters are given by 4(n− 4) + 1

Plucker coordinates, with n frozen variables and 3(n− 5) unfrozen ones. The square

moves still correspond to mutations but only those special ones acting on a node with

two incoming and two outgoing arrows (dual to a square face), and such mutations

generate new letters that are still Plucker coordinates.

2.2 Letters of n = 6 and n = 7 invariants

From the discussions above, one can determine the collection of letters for all possible

Yangian invariants for n = 6. In fact, either from the union of alphabets for NMHV

invariants or from that of N2MHV one, we get 9 + 6 = 15 A coordinates of G(4, 6).

Now we move to n = 7, where in addition to NMHV Yangian invariants, we also

have two classes of N2MHV ones, which are the first cases with letters other than

Plucker coordinates. Since N2MHV invariants are given by parity-conjugate of the

NMHV ones for n = 7, the poles of the former can be obtained as the parity conjugate

of poles of the latter (which are Plucker coordinates). It is straightforward to see

that parity conjugate of 〈ii+1jj+1〉 and 〈i−1ii+1j〉 are proportional to Plucker

coordinates of the same type, thus for n = 7 the only kind of Plucker coordinates

with non-trivial parity conjugate are 〈1346〉 and cyclic permutations; under parity,

we have 〈7(12)(34)(56)〉 etc., and we see that 35 unfrozen cluster variables of G(4, 7)

appear as poles of Yangian invariants. However, we will see new letters appearing

for N2MHV invariants, which are not parity conjugate of Plucker coordinates, and

they are crucial for getting all cluster variables in this case.

A representative plabic graph of a Yangian invariant in the first class is

– 6 –

1

23

4

56

7α3

α2α4

α5

α6

α7

α8

α1

The corresponding C matrix reads(1 α8 α2 + α4 + α6 (α2 + α4 + α6)α7 (α2 + α4)α5 α2α3 0

0 0 1 α7 α5 α3 α1

)and the solution of C(α) · Z = 0 is

α1 =〈3456〉〈4567〉

, α2 = −〈4567〉〈5(12)(34)(67)〉〈2567〉〈3456〉〈3457〉

, α3 = −〈3457〉〈4567〉

, α4 = −〈4567〉〈7(12)(34)(56)〉〈2567〉〈3457〉〈3467〉

,

α5 =〈3467〉〈4567〉

, α6 = −〈1267〉〈4567〉〈2567〉〈3467〉

, α7 = −〈3567〉〈4567〉

, α8 = −〈1567〉〈2567〉

,

where we have defined

〈a(bc)(de)(fg)〉 ≡ 〈abde〉〈acfg〉 − 〈abfg〉〈acde〉 .

Again since we are interested in polynomials of Plucker coordinates, the letters for

this plabic graph are the following 10:

〈1267〉, 〈1567〉, 〈2567〉, 〈3456〉, 〈3457〉, 〈3467〉,〈3567〉, 〈4567〉, 〈5(67)(34)(12)〉, 〈7(56)(34)(12)〉.

(2.3)

By considering cyclic rotations of the labels, we see 7 new letters of the form

〈i (i i+1)(i+2 i+3)(i−2 i−1)〉 for i = 1, · · · , 7, in addition to the 7 + 28 Plucker

coordinates. However, if we compare to the 42 (unfrozen) A coordinates of G(4, 7),

7 variables of the form 〈i(i−1 i+1)(i+2 i+3)(i−3 i−2)〉 (for i = 1, · · · , 7) are still

missing. Can we find such cluster variables as letters of this Yangian invariant?

Very nicely, we will see that by applying square moves, these new letters appear

in equivalent plabic graphs for the same Yangian invariant. Recall that it is more

convenient to describe the square moves using face variables, which are given by

monomials of our coordinates (certain edge variables) [1]. For an internal face with

face variable f , after a square move, the only new factor introduced in the new set

of face or edge variables is given by 1 + f .

– 7 –

We consider square move on either of the two internal faces (the left one is

adjacent to α2, α4, and the right one is adjacent to α4, α6); their face variables are

f1 =α4

α2

=〈3456〉〈7(12)(34)(56)〉〈3467〉〈5(12)(34)(67)〉

, f2 =α6

α4

=〈1267〉〈3457〉〈7(12)(34)(56)〉

.

After performing the square move, we obtain the following new factors,

1 + f1 = −〈3457〉〈6(12)(34)(57)〉〈3467〉〈5(12)(34)(67)〉

, 1 + f2 =〈1257〉〈3467〉〈7(12)(34)(56)〉

, (2.4)

and we see that a new letter 〈6(12)(34)(57)〉 appears. This letter and its cyclic

permutations exactly give the missing 7 cluster variables mentioned above!

A representative plabic graph of a Yangian invariant in the second class is

1

23

4

56

7α3

α2 α4

α5

α6α7

α8

α1

The corresponding C matrix reads(1 α8 α2 + α4 + α7 (α2 + α4)α6 (α2 + α4)α5 α2α3 0

0 0 1 α6 α5 α3 α1

)and the solution of C(α) · Z = 0 is

α1 =〈3456〉〈4567〉

, α2 = −〈4567〉〈3(12)(45)(67)〉〈2367〉〈3456〉〈3457〉

, α3 = −〈3457〉〈4567〉

,

α4 = −〈1237〉〈4567〉〈2367〉〈3457〉

, α5 =〈3467〉〈4567〉

, α6 = −〈3567〉〈4567〉

, α7 =〈1267〉〈2367〉

, α8 = −〈1367〉〈2367〉

.

We obtain the following letters, which also miss the last class of 7 cluster variables:

〈1237〉, 〈1267〉, 〈1367〉, 〈2367〉, 〈3456〉, 〈3457〉, 〈3467〉, 〈3567〉, 〈4567〉, 〈3(12)(45)(67)〉.(2.5)

Similarly we can apply square moves of either of the following two internal faces (the

left is adjacent to α2, α4, and the right one is adjacent to α4, α7) with variables

f1 =α4

α2

=〈1237〉〈3456〉〈3(12)(45)(67)〉

, f2 =α7

α4

= −〈1267〉〈3457〉〈1237〉〈4567〉

,

– 8 –

and the new factors obtained from such moves are given by

1 + f1 =〈1236〉〈3457〉〈3(12)(45)(67)〉

, 1 + f2 = −〈7(12)(36)(45)〉〈1237〉〈4567〉

. (2.6)

Thus we find yet another new letter 〈7(12)(36)(45)〉, and together with cyclic per-

mutations, they give 7 new variables which are not cluster variables! In fact, by

considering all possible plabic graphs for these Yangian invariants, we obtain an al-

phabet that consists of all 42 unfrozen cluster variables of G(4, 7) (plus 7 frozen ones),

as well as 14 new letters that are not cluster variables (all from the second type of in-

variants); 7 in the cyclic class of 〈7(12)(36)(45)〉 and 7 in the class of 〈7(14)(23)(56)〉.These 63 letters make up the complete alphabet for n = 7 Yangian invariants.

2.3 Algebraic letters of N2MHV invariants

Finally, we present our main example of algebraic letters, namely leading singularities

of four-mass boxes (see the left figure below). Without loss of generality, we consider

that of the four-mass box (a, b, c, d) = (1, 3, 5, 7) for n = 8 (the right figure below;

the other four-mass box (a, b, c, d) = (2, 4, 6, 8) is given by cyclic rotation by 1).

a

b−1

b

c−1c

d−1

d

a−1

·· ·

· ··

···

·· ·

1

2 3

4

5

67

8α2 α4

α5

α3

α6 α7

α8

α1

For this representative plabic graph, the corresponding C matrix reads(1 α8 α3 + α6 (α3 + α6)α7 α3α5 α3α4 0 0

0 0 1 α7 α5 α4 α2 α1

)and the solution of C(α) · Z = 0 is

α1 = − 〈3456〉〈127B〉〈456A〉〈128B〉

, α2 =〈3456〉〈456A〉

, α3 = −〈456A〉〈128B〉〈3456〉〈278B〉

, α4 = −〈345A〉〈456A〉

,

α5 =〈346A〉〈456A〉

, α6 =〈1278〉〈278B〉

, α7 = −〈356A〉〈456A〉

, α8 = −〈178B〉〈278B〉

where the two twistors A,B are parametrized as

A = Z7 +α1

α2

Z8 =: Z7 + αZ8, B = Z3 + α7Z4 =: Z3 + βZ4,

– 9 –

then α, β satisfy 〈12AB〉 = 0 and 〈56AB〉 = 0, i.e.

α = −〈5673〉+ 〈5674〉β〈5683〉+ 〈5684〉β

, β = −〈1237〉+ 〈1238〉α〈1247〉+ 〈1248〉α

, (2.7)

and we have two solutions for these quadratic equations, which we denote as X+

and X− for X = A,B. This is the source of letters involving square roots, and the

discriminant of the quadratic equations, ∆, reads

∆ := (〈(34) ∩ (127)568〉+ 〈(34) ∩ (128)567〉)2 − 4〈7(12)(34)(56)〉〈8(12)(34)(56)〉 .

All the algebraic letters for this invariant involve the square root of ∆. Therefore,

we have the following 2 Plucker coordinates and 8 A-like algebraic letters:

〈3456〉, 〈1278〉, 〈456A〉, 〈356A〉, 〈346A〉, 〈345A〉, 〈128B〉, 〈127B〉, 〈178B〉, 〈278B〉 .(2.8)

For each algebraic letter, one can plug in two solutions, X+ and X− but they are

not multiplicative independent since the product is a rational function (which is

expected to be given by rational letters from other Yangian invariants), thus we have

8 independent algebraic letters for this plabic graph.

Quite nicely, by a square move on the internal face, we obtain yet another alge-

braic letter. The face variable of the internal face is

f =α6

α3

= − 〈1278〉〈3456〉〈456A〉〈128B〉

, (2.9)

and the corresponding square move produces a new factor in the numerator:

1 + f =〈456A〉〈128B〉 − 〈1278〉〈3456〉

〈456A〉〈128B〉=〈(AB) ∩ (456)128〉〈456A〉〈128B〉

. (2.10)

Therefore, from the graph and the one after square move, we find 9 algebraic letters:

the 8 in (2.8) and 〈(AB) ∩ (456)128〉 in (2.10), which are all A-like variables. It is

straightforward to check that these 9 algebraic letters generate the same space as the

9 algebraic letters with this ∆ [34], modulo some rational letters.

In fact, we can find all possible algebraic letters for this Yangian invariant by

cyclic symmetry. The four-mass box is invariant under cyclic rotation by 2, and

one can apply it to the 8 algebraic letters in (2.8), which take the form 〈Aijk〉 and

〈Bi′j′k′〉 for i, j, k ∈ {3, 4, 5, 6} and i′, j′, k′ ∈ {7, 8, 1, 2}. Under the rotation, we

obtain the following 8 letters: 〈A′ i+ 2 j + 2 k + 2〉, 〈B′ i′ + 2 j′ + 2 k′ + 2〉 with

A′ = Z1 + α′Z2, B′ = Z5 + β′Z6,

and α′ and β′ are generated by cyclic rotation by 2 of α and β respectively (they

share the same square root). Thus, even without performing square moves explicitly,

– 10 –

we find 16 distinct algebraic letters (if we rotate by 2 again, no new letters appear);

one can check that 9 out of the 16 letters are multiplicative independent.

Alternatively, we can write the algebraic letters similar to cluster X variables,

which are dual conformally invariant. In fact, in the above example, we already see

two DCI letters which are internal face variable f and the one after square move,

1 + f . Recall the well-known variables z and z defined by the equations:

zz =〈1278〉〈3456〉〈1256〉〈3478〉

, (1− z)(1− z) =〈1234〉〈5678〉〈1256〉〈3478〉

. (2.11)

It is easy to see that f = z/(1− z) and 1 + f = 1/(1− z).

To find X -like letters more systematically, let’s consider DCI ratios of the form

f(X+)/f(X−), where f(X) is any of the 16 A-like letters with X = A,B,A′, B′. This

is a natural way to find 16 X -like letters, which are not multiplicative independent

and the rank is 9 as expected. Together with the other half from the box (2, 4, 6, 8),

these X -like letters generate precisely the same space as those DCI algebraic letters

given in [34]! In fact, the new result gives a nice understanding of the multiplicative

relations of [34] and provides a nice basis.

Since this is the only type of n-point N2MHV algebraic Yangian invariants, we

have exhausted algebraic letters for k = 2: for each four-mass box (a, b, c, d), there

are 9 independent algebraic letters which share the same square root.

3 Discussions

In this note, we have proposed an algebraic problem of finding the collection of

letters, or arguments of d log’s, of any Yangian invariant using parameterizations

given by plabic graphs. For a given plabic graph, the 4k polynomial equations

C(α) · Z = 0 provide a map from 4k-dimensional cell of G+(k, n), parametrized by

4k α’s, to a collection of functions defined on G(4, n); the alphabet of a Yangian

invariant consists of such functions for all plabic graphs related to each other by

square moves. In addition to alphabet of NMHV and MHV invariants which are

just Plucker coordinates, we show that for n = 6, 7, the union of alphabets for all

invariants includes all cluster A variables of G(4, n), and for n = 8 the algebraic

letters of N2MHV invariants coincide with the 18 multiplicative-independent symbol

letters found for two-loop NMHV amplitudes [34].

We have only provided some data for the general problem, and it would be highly

desirable to have a general method for constructing letters for any plabic graph,

without the need of case-by-case study. This may involve some algorithm directly in

Z space, which mimics the construction of cells in G+(k, n) from the map C ·Z = 0 [1].

Such a construction would circumvent the bottleneck of our computation, namely the

need to scan through all plabic graphs of a Yangian invariant.

– 11 –

Relatedly, another pressing question here is how to systematically understand

the transformations of letters under square moves of the plabic graphs. As we have

seen in the top-cell (MHV) case, these transformations correspond to certain cluster

transformations, and we would like to understand the analog of these for lower-

dimensional cells. The appearance of non-cluster variables in the alphabet of certain

Yangian invariant is interesting. By studying all possible types of (rational) Yangian

invariants with k = 2 [1], we find that only two of them, the one we have seen

above for n = 7 and one for n = 8, contain such non-cluster variables. It would be

interesting to understand the origin of these non-cluster variables.

Going into fine structures of the alphabet, one can ask “cluster-adjacency” kind

of questions not for the poles of a Yangian invariant, but rather the letters which

appear in the d log form. That is, what letters can appear as arguments of a 4k-

dim d log form? When the letters are all cluster variables, we expect them to be in

the same cluster, but we will need some new ideas when algebraic letters (and those

non-cluster variables) are involved. The answer to this kind of questions may provide

new insights for the study of symbol alphabet of multi-loop amplitudes.

Last but not least, we would like to study this problem at higher n and k. Al-

ready for n = 8, it would be interesting to find the complete alphabet show that the

union of alphabets include all the 180 rational letters of [34]. For n = 9, we find that

the 9× 9 independent algebraic letters from the 9 N2MHV four-mass boxes are not

enough for just the two-loop NMHV amplitudes [35]. This is not surprising since

there are two types of algebraic Yangian invariants for k = 3 (both with Γ(C) = 2),

for which we have not been able to determine the alphabet yet. We leave all these

fascinating open questions to future investigations.

Notes added: During the preparation of the manuscript, [36] appeared on arXiv

which has significant overlap with the result presented in this note.

Acknowledgements

We are grateful to N. Arkani-Hamed for motivating us to study this problem and for

inspiring discussions. We thank Chi Zhang for collaborations on related projects.

All the plabic graphs in this paper are made using the Mathematica package

positroids.m of J. Bourjaily [37]. This work is supported in part by Research

Program of Frontier Sciences of CAS under Grant No. QYZDBSSW-SYS014 and

National Natural Science Foundation of China under Grant No. 11935013.

– 12 –

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